Is algorithmic bias due to the training dataset used?
As the name implies, algorithmic bias is related with the used algorithm. Due to the way it was programmed or devised, the algorithm will be biased in some of its samples.

From Communications of the ACM:


  [Algorithms] often inadvertently pick up the human biases that are incorporated when the algorithm is programmed, or when humans interact with that algorithm.


A machine learning model can be biased if the wrong data-set is used, of course. That is usually only referred to as bias, and often associated with the bias-variance tradeoff.